Thursday, October 20, 2011

Projectile lab

When you throw a ball, the ball tends to be in motion. Such motion is called projectile motion. The purpose of this lab is to study projectile motion and its properties.

Materials required: Clams, Scale, Rounded plastic ball, Spring Gun mounted on table, Paper, Black paper
Method:
The experiment is done to measure the distance that the ball will travel when it is shot from the spring gun.  For this purpose, we mounted a gun on the table and clamp it. We determine the vertical distance from the ground to the gun. We found out that the distance was (1.03±0.02) m. We put paper at some horizontal distance from the table where the gun was leveled. We made sure all the screws and clamps were tight.
a)      Firing at an angle of 0 degree
We adjust the pointer so that it indicates 0 degree. Then we shot the plastic ball from the spring gun. The ball hits the paper. We shot the ball ten times and we measure the distance between the spring gun and the paper. We assume that the initial distance is at x=0. For the experiment we used spring gun but the basic setup of the experiment looks like the figure.




After the ball is fired then the ball is in motion. At that time the force acting on it is gravity (mg where m is the mass of the ball and gravity is the horizontal force).
So, from Newton’s second law of motion
                                                                      Fnet=ma (a is acceleration)
So, we can write,
                                                                      Fx=max
                                                                      Fy=may
So,
                                                                      Fy=-mg
There is no force acting on X-direction. So,
                                                                      F=0
                                                                        i.e. ax=0
So, the only force acting on it is vertical force. To find the final distance it travelled, we used kinematic equations.
For X-direction where there is no acceleration,
                                                                      Final distance(xf)=initial distance(xi) + vxt
                                                                                                Where vx= velocity in x-direction
                                                                                                                t= time taken (initial time is taken as zero)
For Y direction
yf = yi+vyt-0.5gt2

The horizontal and vertical motion is independent of each other except that they have common time. So, the time that we calculate for horizontal motion is the same for vertical motion and vice- versa.

 We consider the vertical motion. Here, yf=0 and yi = h (where h is the distance from the ground to the gun)
Therefore, the equation becomes,
0 t2
(since the launch velocity is zero)
gt2 = 2h
t2 =
t =
so, we can calculate time from this formula which equals 0.459 s.
Now, using t for horizontal motion,
xf =xi+ vx
(x=0 at initial position)
Hence xf= vx
 we can calculate the launch speed using the distance and time

For distance, we shot the ball ten times.
We calculated the distance for ten shots and found the following results:

Shot
Distance(m)
Shot
Distance(m)
1
1.83
6
1.86
2
1.88
7
1.88
3
1.84
8
1.93
4
1.91
9
1.88
5
1.98
10
1.97
To find out the average horizontal distance(range) we took the average from the ten results

Xf=
                                                 Xf=1.896m and the variance is about 0.15m
If we consider the uncertainty then Xf = (1.896±0.15)m.
a)      Firing at an angle of 30 degree
We adjust the pointer so that it indicates 30 degree. Then we shot the plastic ball from the spring gun.  We repeat the same procedure.  We shot the ball ten times and we measure the distance between the spring gun and the paper. Again, we assume that the initial distance is at x=0. The basic setup for the diagram looks like the figure.
 After the ball is fired then the ball is in motion. At that time the force acting on it is gravity (mg where m is the mass of the ball and gravity is the horizontal force).

So, from Newton’s second law of motion
                                                                      Fnet=ma (a is acceleration)
So, we can write,
                                                                      Fx=max
                                                                      Fy=may
So,
                                                                      Fy=-mg
There is no force acting on X-direction. So,
                                                                      F=0
                                                                        i.e. ax=0

Considering the freefall diagram for ball at any point in projectile motion
                                                                                                                      
Here,
xi=0 ; yf = 0
xf = R; yi = h
Velocity of the ball can be splitted into two parts.
Vx= Vcosα
Vy = Vsinα

For X-direction where there is no acceleration,
                                                                       (xf)=xi + Vx.t
                                                                                                Where Vx = velocity in x-direction
                                                                                                                t= time taken (initial time is taken as zero)
For Y direction
yf = yi+ Vy .t-0.5gt2

The horizontal and vertical motion is independent of each other except that they have common time. So, the time that we calculate for horizontal motion is the same for vertical motion and vice- versa.

 We consider the vertical motion. Here, yf=0 and yi = h (where h is the distance from the ground to the gun)
Therefore, the equation becomes,
0 t2
t2  =0
-(1/2)g.t2+ Vy.t+h =0
The equation is identical with the equation ax2+bx+c=0 . Hence using quadratic equation formula,
Comparing the equation obtained for t with x we get,
a = -(1/2)g , b= Vy, , c=h

Therefore we can write,
Now, using t for horizontal motion,
So, from this we came to know that distance(range) is related with the angle at which we shot. Hence, by using the distance and putting the values of angle α, gravity g, height h, we can find the velocity at which the ball was shot.

We calculated the distance for ten shots and found the following results:

Shot
Distance(m)
Shot
Distance(m)
1
2.58
6
2.53
2
2.66
7
2.61
3
2.61
8
2.66
4
2.61
9
2.64
5
2.51
10
2.54

To find out the average horizontal distance(range) we took the average from the ten results

Xf=
                                                 Xf=2.595m and the variance is about 0.15m
If we consider the uncertainty then Xf = (2.595±0.15)m.

Conclusion:
So, basically the experiment was done to determine the launch velocity and we came to conclude that it is affected by the angle at which we shot the ball or any other projectile motion.
We did not account for air resistance acting on the ball. Instead, we neglected it. In the formulas stated above, there is no relation of velocity with the mass. So, basically, it can be said that the mass of the projected object does not contribute to the velocity of the ball. However, I think that the launch velocity is affected by mass though I don’t have any valid reasons to prove it.